Worst-case hardness suffices for derandomization: a new method for hardness-randomness trade-offs

Chapter 8 Notes and References

“…The fastest known deterministic algorithms for approximately counting the number of satisfying assignments to a DNF formula are from [280] and [178] (depending on whether the approximation is relative or additive, and the magnitude of the error). The fact that hitting set generators imply BPP = P (Problem 7.8) was first proven by Andreev, Clementi, and Rolim [27]; for a more direct proof, see [173]. Problem 7.9 (that PRGs vs. uniform algorithms imply average-case derandomization) is from [216].…”

“…For example: Definition 2. 27. prBPP is the class of promise problems Π for which there exists a probabilistic polynomial-time algorithm A such that…”

“…27. That is, we define a neighbor function Γ : 9) where f (Y ) is a polynomial of degree at most n − 1 over F q , and we define This is possible because…”

Extracting all the randomness and reducing the error in Trevisan's extractors

“…This is the starting point for our work. Explicit constructions of extractors and dispersers have a wide variety of applications, including simulating randomized algorithms with weak random sources [Zuc96]; constructing oblivious samplers [Zuc97]; constructive leader election [Zuc97,RZ98]; randomness-efficient error reduction in randomized algorithms and interactive proofs [Zuc97]; explicit constructions of expander graphs, superconcentrators, and sorting networks [WZ99]; hardness of approximation [Zuc96,Uma99]; pseudorandom generators for space-bounded computation [NZ96,RR99]; derandomizing BPP under circuit complexity assumptions [ACR97,STV99]; and other problems in complexity theory [Sip88,GZ97].…”

Uniform derandomization from pathetic lower bounds

“…The main thread of this line of research dates back to the work of Shamir [4], Yao [5] and Blum & Micali [6], and involves showing that, given a suitably hard function f , one can construct pseudorandom generators and hitting-set generators. Much of the progress on this front over the years has involved showing how to weaken the hardness assumption on f and still obtain useful derandomizations [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. In rare instances, it has been possible to obtain unconditional derandomizations using this framework; Nisan [23], Nisan & Wigderson [24] and Viola [25] showed that uniform families of probabilistic AC 0 circuits can be simulated by uniform deterministic AC 0 circuits of size n log O (1) n .…”